9 METRIZABLE GROUPS .41 Hence the property of being a Cauchy sequence with respect to some left- invariant distance is independent of the choice of distance, and depends only on the topology of G. We shall say simply that such a sequence is a left Cauchy sequence in G. A right Cauchy sequence in G is defined analogously by replacing x~*xm in (12.9.2) by xnx~l. It can happen that a left Cauchy sequence is not a right Cauchy sequence (Problem 8); but if (jcj is a left Cauchy sequence, then (x"1) is a right Cauchy sequence. Since the mapping x\-+x~l is continuous it follows that, if every left Cauchy sequence in G converges, then so does every right Cauchy sequence. In this case G is said to be a complete metrizable group: it is complete with respect to every left- invariant distance and every right-invariant distance defining the topology ofG. (12.9.3) Let G, G' be two metrizable groups. Then every continuous homo- morphism f of G into G' is uniformly continuous with respect to left- (resp. right-) invariant distances on G and G7. Let d, d' be two such distances on G, G', respectively. By hypothesis, for each e > 0 there exists d > 0 such that the relation d(e, z) ^ 5 implies d'(e',f(z))£e. Hence if d(x,y) = d(e, x~ly) £ 6, then d'(f(x\ f(y)) £ e, because d'(f(x),f(y)) = d'(S, (/Mr VO)) = d'(e',f(x-*y)-) since /is a homomorphism. (12.9.4) Let Gl5 G2 be two metrizable groups such that G2 is complete, and let H! (resp. H2) be a dense subgroup ofGt (resp. G2). Then every continuous homomorphism u : Hj -> H2 can be extended uniquely to a continuous homo- morphism u : Gi ~~> G2 . If Gx is also complete and if u is an isomorphism (of topological groups) of Hj onto H2 , then u is an isomorphism (of topological groups) of G! onto G2 . There exist left-invariant distances on Gl and G2 , and the existence of the continuous extension u then follows from (12.9.3) and (3. 15.6). The fact that u is a group homomorphism follows from the principle of extension of identities applied to the two functions (jc, y) i-> u(xy) and (x, y)\-+u(x)u(y) on Gl x Gx (having regard to (3.20.3)). Finally, if Gx is complete and ifv : H2 -» Hj is the inverse of the isomorphism u, then v can be extended to a continuous homo- morphism v of G2 into Gx . Since v ° u and u o v agree with the identity mappings on Ht and H2 respectively, they are the identity mappings on G! and G2 respectively, by virtue of the principle of extension of identities and (3.11.5). This completes the proof. the